Model comparison using rjmcmc & gamma prior on B
Bayes Factor of compared models determined. With alpha transform.
Contents
- Part I: RJMCMC - \(exp(1)\) prior on \(\beta\)
- Data
- Data simulated using the Super Spreading Events model \((\alpha, \beta, \gamma)\). Four datasets simulated - small & large outbreaks.
- Data simulated using the Baseline model of a regular events spreading model \((\alpha)\). Two datasets simulated.
- RJMCMC
- RJMCMC implemented, jumping between the SSE and baseline model.
- Bayes Factor between the two models calculated
- Part II: RJMCMC - \(\Gamma(k, \theta)\) prior on \(\beta\)
- The steps are repeated as in part I with the exception of a \(\Gamma(k, \theta)\) prior on \(\beta\). Five different gamma priors were trialled including a including a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 2)\), \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 3, \hspace{1mm} 3)\), \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 4, \hspace{1mm} 4)\) and \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 5, \hspace{1mm} 5)\)
- Identical datasets are used
Background
RJMCMC was implemented whereby the algorithm jumped between the Baseline model \(M_1\) of an epidemic with regular spreading events \((\alpha)\) and the Super Spreading Events (SSE) model \((\alpha, \beta, \gamma)\) which has both regular spreading events with rate \(\alpha\) and super spreading events with rate \(\beta\) and multiplicative factor \(\gamma\).
Bayes Factor
Bayesian model comparison is a method of model selection based on Bayes factors. The aim of the Bayes factor is to quantify the support for one model over another, e.g model \(M_1\) over model \(M_2\). The Bayes Factor BF is as follows;
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)P(D)}{P(M_1)}}{\dfrac{P(M_2|D)P(D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_2 | D)}
\]
when \(P(M_1) == P(M_2)\), otherwise
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)}{P(M_1)}}{\dfrac{P(M_2|D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_1)} \cdot \dfrac{P(M_2)}{P(M_2|D)}
\]
where \(P(D|M_1)\) is the model evidence, specifically the marginal likelihood integrand;
\[ P(D|M_1) = \int P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta) \hspace{1 mm} P(\hspace{1 mm}\theta \hspace{1 mm}| M_1) \hspace{1 mm}d \theta\] and the first term in the integrand \(P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta)\) is the likelihood and the second term \(P(\hspace{1 mm}\theta \hspace{1 mm}| M_1)\) is the prior on the model parameter \(\theta\).
Interpretation of the Bayes Factor results
A Bayes Factor > 1 signifies that \(M_1\) is more strongly supported by the data under consideration than \(M_2\). Harold Jefferys gave a scale of interpretation of the Bayes Factor;
| Bayes Factor | Bayes Factor equivalence | Evidence Strength |
|---|---|---|
| < 10^0 | < 1 | Negative (supports M_2) |
| [10^0, 10^1/2] | [1, 3.16] | Weak evidence |
| [10^1/2, 10^1] | [3.16, 10] | Substantial |
| [10^1, 10^3/2] | [10, 31.62] | Strong |
| [10^3/2, 10^2] | [31.62, 100] | Very strong |
| > 10^2 | > 100 | Decisive |
Part I: RJMCMC between Base model & SSE Model - exp(1) prior used for beta
An \(exp\hspace{1mm}(\beta; \hspace{1mm}1)\) has density;
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.32 0.1 0.01 10 0.18 1.8 0.23 70.6 86.57
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 941 12.33 134 99.45 1081 843
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 244 77.55 842 8070 9.45 0.8913
## beta_pc_non_0 bf
## 1 0.1087 8.2
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.66 0.1 0.05 10 4.36 1.8 0.91 1.94 4.35
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 435 1.8 180 4.2 420 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.45 0.1 0.15 10 6.53 1.8 1.4 5.2 6.16
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 616 2.01 201 6.15 615 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.78 0.1 0.02 10 0.26 1.8 0.82 10.43 83.89
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1286 14.29 219 93.41 1432 1435
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 98 93.61 1434 7032 16.94 0.8467
## beta_pc_non_0 bf
## 1 0.1533 5.523
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.75 0.1 0.02 10 0.18 1.8 1.77 3.66 54.12
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 657 8.81 107 42.5 516 1202
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 12 99.01 1201 7584 13.67 0.8786
## beta_pc_non_0 bf
## 1 0.1214 7.237
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.73 0.1 0.08 10 0.45 1.8 1.85 10.17 62.97
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1779 9.98 282 40.25 1137 2600
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 225 92.04 2599 4575 36.23 0.7175
## beta_pc_non_0 bf
## 1 0.2825 2.54Part 2: Gamma Prior on Beta
A gamma prior, \(\Gamma(\beta; k, \theta)\) on beta was also trialed whereby \(k\) determines the shape of the distribution and \(\theta\) governs the scale. The gamma distribution function is as follows;
\[ \Gamma(\beta; k, \theta) = \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \]
A range of gamma priors on beta are used including a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 2)\), \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 3, \hspace{1mm} 3)\), \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 4, \hspace{1mm} 4)\) and \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 5, \hspace{1mm} 5)\). Each have a mean of \(k \cdot \theta\). A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), is as follows;
A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 2)\);
And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 3)\);
And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}4, \hspace{1mm} 4)\);
And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}5, \hspace{1mm} 5)\);
In the Metropolis acceptance step, the logs of all quantities are determined and evaluating \(log\hspace{1mm}( \Gamma\hspace{1mm}(\beta; \hspace{1mm} k, \hspace{1mm} \theta))\) gives;
\[log\bigg( \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \bigg)\]
\[ = \dfrac{1}{log\Gamma(k)\cdot klog(\theta)} \cdot (k-1) \cdot log(\beta) \cdot \dfrac{-\beta}{\theta} \]
Gamma Prior \(\Gamma\)(\(\beta\); 2, 2.5)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.38 0.1 0.04 10 0.33 1.8 0.47 71.41 0
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 0 16.37 292 0.22 4 1330
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 454 74.55 1329 6886 16.18 0.8216
## beta_pc_non_0 bf
## 1 0.1784 4.605
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.16 0.1 0.5 10 3.74 1.8 2.02 55.09 0
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 0 14.81 1481 0 0 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.07 0.1 0.5 10 3.73 1.8 1.93 25.29 0
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 0 3.54 354 0 0 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.82 0.1 0.03 10 0.28 1.8 0.87 10.82 0.06
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1 13.55 212 0.77 12 1522
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 42 97.31 1522 6913 18.04 0.8435
## beta_pc_non_0 bf
## 1 0.1565 5.39
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.75 0.1 0.02 10 0.19 1.8 1.78 3.29 0
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 0 9.82 119 0.41 5 1206
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 6 99.5 1205 7582 13.71 0.8788
## beta_pc_non_0 bf
## 1 0.1212 7.251
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.73 0.1 0.08 10 0.45 1.8 1.85 9.44 0.18
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 5 10.16 282 2.27 63 2577
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 199 92.83 2577 4646 35.68 0.7223
## beta_pc_non_0 bf
## 1 0.2777 2.601
Gamma Prior \(\Gamma(\beta; 3, 2)\)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.32 0.1 0.02 10 0.26 1.8 0.32 71.3 95.7
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1335 13.05 182 88.03 1228 1026
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 369 73.55 1025 7579 11.91 0.8605
## beta_pc_non_0 bf
## 1 0.1395 6.168
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.53 0.1 0.08 10 4.91 1.8 0.96 3.37 7.05
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 705 3.24 324 6.79 679 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.59 0.1 0.12 10 6.1 1.8 1.36 4.36 4.18
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 418 1.63 163 4.14 414 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.81 0.1 0.03 10 0.26 1.8 0.85 10.31 90.29
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1395 13.4 207 79.48 1228 1438
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 107 93.07 1437 7017 17 0.8455
## beta_pc_non_0 bf
## 1 0.1545 5.472
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.76 0.1 0.02 10 0.19 1.8 1.78 3.73 55.3
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 689 9.63 120 36.76 458 1232
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 14 98.88 1231 7522 14.06 0.8754
## beta_pc_non_0 bf
## 1 0.1246 7.026
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.73 0.1 0.09 10 0.48 1.8 1.87 10.16 61.81
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1772 9.7 278 31.88 914 2608
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 259 90.97 2607 4525 36.55 0.7133
## beta_pc_non_0 bf
## 1 0.2867 2.488
Gamma Prior \(\Gamma(\beta; 3, 3)\)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.32 0.1 0.02 10 0.26 1.8 0.32 71.3 95.7
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1335 13.05 182 88.03 1228 1026
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 369 73.55 1025 7579 11.91 0.8605
## beta_pc_non_0 bf
## 1 0.1395 6.168
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.53 0.1 0.08 10 4.91 1.8 0.96 3.37 7.05
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 705 3.24 324 6.79 679 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.59 0.1 0.12 10 6.1 1.8 1.36 4.36 4.18
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 418 1.63 163 4.14 414 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.81 0.1 0.03 10 0.26 1.8 0.85 10.31 90.29
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1395 13.4 207 79.48 1228 1438
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 107 93.07 1437 7017 17 0.8455
## beta_pc_non_0 bf
## 1 0.1545 5.472
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.76 0.1 0.02 10 0.19 1.8 1.78 3.73 55.3
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 689 9.63 120 36.76 458 1232
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 14 98.88 1231 7522 14.06 0.8754
## beta_pc_non_0 bf
## 1 0.1246 7.026
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.73 0.1 0.09 10 0.48 1.8 1.87 10.16 61.81
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1772 9.7 278 31.88 914 2608
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 259 90.97 2607 4525 36.55 0.7133
## beta_pc_non_0 bf
## 1 0.2867 2.488
Gamma Prior \(\Gamma(\beta; 4, 4)\)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.31 0.1 0.02 10 0.27 1.8 0.34 71.28 95.47
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1390 13.6 198 86.68 1262 1077
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 379 73.97 1076 7467 12.6 0.8544
## beta_pc_non_0 bf
## 1 0.1456 5.868
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.73 0.1 0.07 10 5.08 1.8 1.15 2.62 6.05
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 605 2.74 274 5.84 584 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.55 0.1 0.14 10 7.25 1.8 1.53 7.51 7.71
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 771 3 300 8.09 809 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.79 0.1 0.02 10 0.27 1.8 0.83 10.55 91.21
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1411 13.57 210 76.08 1177 1444
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 103 93.34 1443 7009 17.07 0.8453
## beta_pc_non_0 bf
## 1 0.1547 5.464
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.76 0.1 0.02 10 0.19 1.8 1.79 3.73 53.71
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 680 9.72 123 35.23 446 1256
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 10 99.21 1255 7478 14.37 0.8734
## beta_pc_non_0 bf
## 1 0.1266 6.899
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.72 0.1 0.09 10 0.48 1.8 1.87 10.05 61.86
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1766 9.42 269 31.38 896 2576
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 279 90.23 2575 4569 36.04 0.7145
## beta_pc_non_0 bf
## 1 0.2855 2.503
Gamma Prior \(\Gamma(\beta; 5, 5)\)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.32 0.1 0.02 10 0.24 1.8 0.31 71.28 95.98
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1314 12.49 171 88.24 1208 1019
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 350 74.43 1018 7612 11.8 0.8631
## beta_pc_non_0 bf
## 1 0.1369 6.305
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.53 0.1 0.08 10 4.91 1.8 0.96 3.37 7.05
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 705 3.24 324 6.79 679 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.59 0.1 0.12 10 6.1 1.8 1.36 4.74 4.99
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 499 1.79 179 4.9 490 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.8 0.1 0.03 10 0.27 1.8 0.85 10.5 90.63
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1422 13.19 207 77.12 1210 1477
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 92 94.14 1477 6953 17.52 0.843
## beta_pc_non_0 bf
## 1 0.157 5.369
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.76 0.1 0.02 10 0.2 1.8 1.78 3.85 54.43
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 682 9.34 117 36.55 458 1241
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 12 99.04 1240 7506 14.18 0.8747
## beta_pc_non_0 bf
## 1 0.1253 6.981
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.72 0.1 0.1 10 0.5 1.8 1.88 10.39 59.66
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1739 9.26 270 31.94 931 2599
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 316 89.16 2598 4486 36.67 0.7085
## beta_pc_non_0 bf
## 1 0.2915 2.431
Bayes Factor - Summary of Results
In this particular setting in which the Baseline model \(\alpha\) and SSE model \((\alpha, \beta, \gamma)\) are compared, the Bayes factor is calculated as;
\[ \dfrac{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples == 0}{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples != 0} \]
The following table summaries the results of the RJMCMC iterations for a number of datasets when both a \(exp(\beta, 1)\) prior and a variety of \(\Gamma(\beta;)\) priors on beta were used.
| Epidemic Data | Max daily infection count | Bayes Factor exp(beta; 1) | Bayes Factor,gamma(2, 2.5) | Bayes Factor,gamma(3, 2) | Bayes Factor,gamma(3, 3) | Bayes Factor,gamma(4, 4) | Bayes Factor,gamma(5, 5) |
|---|---|---|---|---|---|---|---|
| SSE - dies out | 2 | 8.2 | 4.605 | 6.168 | 6.168 | 5.868 | 6.305 |
| SSE spreads | 55 | 0 | 0 | 0 | 0 | 0 | 0 |
| SS spreads | 340 | 0 | 0 | 0 | 0 | 0 | 0 |
| SSE - dies out | 2 | 5.523 | 5.39 | 5.472 | 5.472 | 5.464 | 5.369 |
| Base - spreads | 65 | 7.237 | 7.251 | 7.026 | 7.026 | 6.899 | 6.981 |
| Base - spreads | 45 | 2.54 | 2.601 | 2.488 | 2.488 | 2.503 | 2.431 |